# Multi-body dynamics

The multi-body dynamics (sometimes also called “dynamics of the multi-body systems” (DMKS)) considers the movement process of several (e.g. through joints) constrained bodies of a multi-body system , whereby inertial forces are decisive.

## Research areas

• Numerical simulation
• Stability of movement
• Sensitivity of the movement process with regard to geometric sizes, material sizes and initial conditions
• Determination of the initial configuration
• Optimization of the movement process (speed, energy efficiency, ...)
• inverse movement (backwards in time)
• Control of moving systems

Multi-body systems are divided into rigid-body systems, s. Multi-body system , and flexible multi-body systems . An example illustrates the principle of a multi-body system

## Simulation of multi-body systems

To simulate multi-body systems (for examples see link below), the equations of motion for certain initial conditions (i.e. initial configuration and initial speeds ) must be solved over a certain period of time.

### Independent coordinates

If the equations of motion are described exclusively with independent (non-redundant) coordinates, then solution methods for ordinary differential equations can be used, e.g. B. many types of Runge-Kutta processes or multi-step processes .

### Rigid vs. flexible body

If, furthermore, it is a question of pure rigid body systems, explicit time integration methods can be used efficiently. However, if flexible bodies are included, then special implicit time integration methods (Newmark, Gauss, Radau, Lobatto) are often advantageous because the time steps are not subject to any restrictions in terms of size, while in the case of explicit methods, the time step size must be limited to the size of the highest occurring frequency .

### Redundant coordinates

If in the equations of motion constraints occur numerical solution methods for ordinary differential equations can not readily be used as it is in the equations to differential equations Algebraic - (DAE differential algebraic equations is). DAEs are mainly characterized using the index, which indicates how often the algebraic equations (constraints) have to be differentiated in order to obtain a system of ordinary differential equations. The equations of motion usually have the index 3 for displacement constraints.

There are only a few methods which, with a few modifications (e.g. scaling the equations) and then only limited to differential-algebraic equations with index, can be used to obtain a good approximate solution (e.g. HHT, RadauIIA from 2 levels) .

Index reduction

Mostly, however, a so-called index reduction is used in order to be able to use simpler solution methods. The index reduction takes place by means of the derivation of the constraint with respect to time, whereby constraints in the speeds are obtained from simple displacement constraints. Efficient solution methods for index 2 systems are e.g. B. BDF ( backward difference ) or implicit center point rule, trapezoidal rule or the Newmark method.

drift

By deriving the constraints, these conditions are only met exactly ( machine accuracy ) in the speeds in each time step , but an error in the positions develops over time ( drift ). This error can be reduced or eliminated by stabilization methods. Common stabilization methods are Baumgarte stabilization or Gear Gupta Leimkuhler (GGL) stabilization. The drift with an index 2 formulation can be kept small through very precise integration, but it usually increases linearly. In order to be able to use explicit solution methods, the index has to be reduced to 1, whereby the drift becomes very large and stabilization methods are inevitable.

## literature

• J. Wittenburg: Dynamics of Systems of Rigid Bodies. Teubner, Stuttgart (1977).
• K. Magnus: Dynamics of multibody systems. Springer Verlag, Berlin (1978).
• EJ Haug: Computer-Aided Kinematics and Dynamics of Mechanical Systems. Allyn and Bacon, Boston (1989).
• E. Hairer and Ch. Lubich, and M. Roche: The numerical solution of differential-algebraic systems by Runge-Kutta methods. Lecture Notes in Math. 1409, Springer-Verlag, (1989).
• E. Hairer and G. Wanner: Solving ordinary differential equations II, stiff and differential-algebraic problems. Springer Verlag: Berlin Heidelberg, 1991.
• KE Brenan, SL Campbell, and LR Petzold: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. SIAM, Philadelphia, 1996.
• AA Shabana: Dynamics of multibody systems. Second Edition, John Wiley & Sons (1998).